# Ma 108C. Complex Analysis

### Schedule

MWF 11:00 - 11:55 am, 269 Lauritsen Physics Laboratory

### Office Hours

MW 4:15 - 5:15 pm, math building

In this course, we explore the foundations of complex analysis in one variable. The main object of study are holomorphic functions, which are functions that are differentiable in the complex sense. Strangely enough, if such a function can be differentiated once, it can be differentiated infinitely many times, and in fact admits a power series expansion (i.e. is analytic). To prove this remarkable fact, we develop integration along contours (Cauchy's theorem). Another remarkable property of holomorphic functions is that the value of a holomorphic function at a point z in a domain Omega can be recovered from the values of the function on the boundary of Omega. Further topics include: the Schwarz lemma, the argument principle and residue calculus.

### References

There are many wonderful books on complex analysis. You should acquire one of them:
• Cartan. Elementary Theory of Analytic Functions of One or Several Complex Variables
• Ahlfors. Complex analysis
• Stein, Shakarchi. Complex analysis

Jack Tao

Assignment 1

Assignment 2

Assignment 3

Assignment 4

Practice Exam

Solution 1

Solution 2

Solution 3

Solution 4

### Marking scheme

We will have four homework assignments, worth 15% each. The final will be 40%.